Bellwork

• Classify the following angles based on their measure– 102o

– 37o

• Find the complement and supplement of XYZ if m XYZ=80

• If XY=YZ, is Y the midpoint of XZ? If not, give a counter example.

Clickers

Bellwork Solution

• Classify the following angles based on their measure– 102o

– 37o

Obtuse

Acute

Bellwork Solution

• Find the complement and supplement of XYZ if mXYZ=80

108090 10080180

Bellwork Solution

• If XY=YZ, is Y the midpoint of XZ? If not, give a counter example.

Analyze Conditional Statements

Section 2.2

The Concept• Today we’re going to revisit a small topic from

algebra. • This topic is important to us because it

develops a format for us to use our understanding of inductive reasoning

Definition• Conditional Statement

– Logical statement that uses a hypothesis and a conclusion

Example or IllustrationWritten DefinitionSection & Page Number

Word or Concept

Example or IllustrationWritten DefinitionSection & Page Number

Word or Concept

If-Then statements• Most conditional statements follow an if-then format

• The if part of the statement is the hypothesis and the then portion is the conclusion

If you took good notes, then the test was easier

Hypothesis Conclusion

Translating into if-then formThe ability to translate a statement into if-then form depends

on the ability to see the two statements within one context

For instance

All whales are mammals.

If an animal is a whale, then it is a mammal.

QuizWhich of the following is the correct conditional statement

based on the following regular statement

The weather has to be cold for it to snow.

A. If it is snowing, then the weather is cold.B. If it is cold, then it can snow.C. It will snow, if the weather is cold.

More Definitions• Negation

– The opposite of the original statement

• Converse– A rewritten conditional statement in which the hypothesis and

conclusion are exchanged

• Inverse– A rewritten conditional statement in which both the hypothesis

and conclusion are negated

• Contrapositive– A rewritten conditional statement in which both parts of the

converse are negated

Examples• Conditional Statement

– p→q– If you are a UA student, then you are wearing a polo shirt

• Converse– q→p– If you are wearing a polo shirt, then you are a UA student

• Inverse– ~p→~q– If you are not a UA student, then you are not wearing a polo shirt

• Contrapositive– ~q→~p– If you are not wearing a polo shirt, then you are not a UA student

Examples

The following statement is which kind of statement based on the conditional, “If the cafeteria serves pizza, then the majority of students will eat it.”

If the majority of student’s don’t eat it, then the cafeteria didn’t serve pizza

A. ConverseB. InverseC. Contrapositive

Examples

The following statement is which kind of statement based on the conditional, “If the cafeteria serves pizza, then the majority of students will eat it.”

If the majority of student’s eat it, then the cafeteria served pizza

A. ConverseB. InverseC. Contrapositive

Examples

The following statement is which kind of statement based on the conditional, “If the cafeteria serves pizza, then the majority of students will eat it.”

If the cafeteria didn’t serve pizza, then the majority of student’s didn’t eat.

A. ConverseB. InverseC. Contrapositive

Testing Validity• Statement: Soccer players are athletes• Conditional Statement

– If p, then q– If you are a soccer player, then you are an athlete

• Converse– If q, then p– If you are an athlete, then you are a soccer player

• Inverse– If not p, then not q– If you are not a soccer player, then you are not an athlete

• Contrapositive– If not q, then not p– If you are not an athlete, then you are not a soccer player

A. TrueB. False

Equivalent Statements• More often than not, we see a pattern develop with

conditional statements– Conditional;True– Inverse; False– Converse; False– Contrapositive;True

• In these situations, the conditional and contrapositive are called equivalent statements– The inverse and converse are also

• This is important as it relates to writing definitions

Equivalent Statements• For example• If two lines intersect to form a right angle, then they are

perpendicular lines– The contrapositive is also true– If two lines are not perpendicular, then they do not intersect to

form a right angle

• Can we make a similar statement to define our understanding of parallel lines and how they intersect?– If two lines do not intersect, then they are parallel– Contrapositive: If two lines are not parallel, then they do

intersect

Example• What statement can we make about this picture

1 2

m1+ m2=180o

Bi-Conditional StatementsDoes something special happen when both the

conditional and its converse are true?

Definition: Bi-Conditional Statement: Special conditional statement possible when both the conditional and the converse are true; denote by the phrase, “if and only if”

Bi-Conditional StatementsFor example:

Converse: If plastic bottle is a #1 or #2, then it can be recycled in Kansas City

Conditional: If a plastic can be recycled in Kansas City, then it must be a #1 or #2

T

T

Bi-Conditional: A plastic can be recycled in Kansas City, if and only if it is a #1 or #2

Examples• What could we say about our previous example?

1 2

m1+ m2=180o

On your own• Can the following conditional statement be transformed

into a bi-conditional statement.

If you can see outside, you will see the sunshine

A. YesB. No

On your own• Can the following conditional statement be transformed

into a bi-conditional statement.

If you’re not going to eat your vegetables, then you’re not going to grow up to be big and strong

A. YesB. No

Homework

• 2.2– 1-21, 25, 31, 32

– 4th hour: 1-6, 8-20 even, 25, 31

Most Important Points• Conditional Statements• Negations• Converse• Inverse• Contrapositive